Quantitative research interviews at top trading firms like SIG (Susquehanna International Group) and Jane Street are renowned for their challenging and insightful problem-solving questions. These questions are designed to probe a candidate’s mathematical intuition, logical reasoning, and ability to communicate sophisticated concepts clearly. In this article, we will delve into several real quant research interview questions asked at SIG and Jane Street, solve them in detail, and explain all underlying concepts. Whether you're preparing for your quant interview or simply interested in advanced probability and combinatorics, this guide will equip you with both the solutions and the reasoning techniques used by successful candidates.

Quant Research Interview Questions – SIG and Jane Street

1. Probability in a Colored 3x3x3 Cube

Problem Statement

Suppose you are given a 3x3x3 cube. All 6 faces are colored green. The cube is then cut into 27 smaller cubes. You randomly pick one small cube and see one of its faces is green. What is the probability that the cube is an edge cube? (Susquehanna International Group)

Analysis

Let’s define the types of small cubes:

• Corner cubes: 8 cubes, each has 3 faces green.
• Edge cubes (not corners): Each edge has 1 center cube (excluding corners), 12 edges total, so 12 cubes. Each has 2 green faces.
• Face center cubes (not edge nor corner): Each face has 1 center (not on any edge or corner), 6 faces, so 6 cubes. Each has 1 green face.
• Internal cube: 1 cube in the center, 0 green faces.

Total Small Cubes: 27

We only consider cubes with at least one green face (exclude the internal center cube).

Probability Calculations

Suppose you pick a cube and see a green face. What is the probability it is an edge cube?

Let’s use Bayes theorem. Let \( E \) = “cube is edge cube,” \( G \) = “cube with a green face is observed.”

We compute \( P(E | G) = \frac{P(G | E) P(E)}{P(G)} \).

But since you pick a cube and see a green face, we assume:

• You randomly select one of the 27 small cubes.
• For that cube, you randomly select one of its faces.
• If the face is green, you observe it.
• What is the probability that the cube is an edge cube, given that the observed face is green?

Count the Total Number of Green Faces

• Corner cubes: 8 cubes, each with 3 green faces ⇒ 8 × 3 = 24 green faces
• Edge cubes: 12 cubes, each with 2 green faces ⇒ 12 × 2 = 24 green faces
• Face centers: 6 cubes, each with 1 green face ⇒ 6 × 1 = 6 green faces
• Internal cube: 1 cube, 0 green faces

Total green faces = 24 + 24 + 6 = 54.

Probability the Green Face Comes from an Edge Cube

Number of green faces on edge cubes = 24.
Total green faces = 54.
So, probability = \( \frac{24}{54} = \frac{4}{9} \).

Therefore, the answer is \( \frac{4}{9} \).

Key Concepts

• Careful enumeration of combinatorial cases (corners, edges, centers, internal).
• Application of conditional probability and Bayes’ theorem.
• Uniform probability over faces, not just cubes.

2. Game with Two Players Throwing Two Dice Each

Problem Statement

A game where two players each throw two dice. If any one die matches the other (i.e., if any die from one player matches any die from the other), you win a dollar. If there is no match, you lose a dollar. Would you play this game or not and why? (Susquehanna International Group)

Analysis

The question is: what is the expected value per round?

Total Possible Outcomes

• Each player rolls two dice: 6 × 6 = 36 possible outcomes per player.
• Total combinations: 36 × 36 = 1296.

Number of Winning Outcomes

We want the probability that, among the four dice, at least one die from player 1 matches one from player 2.

Let’s compute the probability there is no match at all, and then subtract from 1.

• Let’s fix player 1’s dice: say they are \( a \) and \( b \) (possibly equal).
• Player 2’s dice: \( c \) and \( d \).
• We want \( c \neq a, b \) and \( d \neq a, b \).
• If \( a = b \): only one number forbidden.
• If \( a \neq b \): two numbers forbidden.

Let’s compute for both cases:

Case 1: \( a = b \)

There are 6 such pairs for player 1 (\( a = b \)), each with probability \( \frac{6}{36} = \frac{1}{6} \).
For each, player 2’s dice must both be not equal to \( a \), so each die has 5 options: 5 × 5 = 25.
So, for this case: \( 6 \times 25 = 150 \) outcomes.

Case 2: \( a \neq b \)

Number of such pairs: \( 6 \) choices for \( a \), \( 5 \) for \( b \), but order doesn't matter (since dice are distinguishable), so 6 × 5 = 30.
Each die must not be \( a \) or \( b \): so each has 4 choices. So, \( 4 \times 4 = 16 \).
So, total: \( 30 \times 16 = 480 \).

Total Number of "No Match" Outcomes

Total = 150 + 480 = 630.

Thus, probability of no match = \( \frac{630}{1296} = \frac{35}{72} \).

So, probability of at least one match = \( 1 - \frac{35}{72} = \frac{37}{72} \).

Expected Value Per Game

• Win $1 with probability \( \frac{37}{72} \)
• Lose $1 with probability \( \frac{35}{72} \)

Expected value per game: \[ \frac{37}{72} \times 1 + \frac{35}{72} \times (-1) = \frac{37 - 35}{72} = \frac{2}{72} = \frac{1}{36} \]

Since the expected value is positive, you should play the game.

Key Concepts

• Counting distinguishable and indistinguishable cases.
• Subtracting complementary probability.
• Expected value calculation for betting games.

3. Probability that Three Random Points on a Unit Circle Form a Triangle Containing the Center

Problem Statement

What is the probability that three random points on a unit circle form a triangle that includes the center of the circle? (AKUNA Capital)

Conceptual Approach

Three random points on the circle define a triangle. For the triangle to contain the center, all the points must not fall within any semicircle. (Otherwise, the triangle does not cover the center.)

Mathematical Explanation

• For the triangle to contain the center, all the points must not be contained in a semicircle.
• Conversely, if all points are within a semicircle, the triangle does not contain the center.

Computing the Probability

Let’s fix one point arbitrarily (say, at 0°), and consider the locations of the other two points.

• Let the circle be parameterized from 0 to 1.
• The other two points are at positions \( x, y \in [0,1) \).
• The points are distributed independently and uniformly.

After sorting the three points on the circle, the triangle contains the center if and only if no arc between consecutive points is greater than half the circle (i.e., \( > 180^\circ \)).

Let’s define the arc lengths between the sorted points as \( a, b, c \), with \( a + b + c = 1 \).
The max arc must be less than \( \frac{1}{2} \).

The probability all arcs are less than \( \frac{1}{2} \) for three random points is \( \frac{1}{4} \).

Alternative Intuitive Solution

For each arrangement, the largest arc length must be less than \( \frac{1}{2} \). The probability that no arc is greater than \( \frac{1}{2} \) is \( \frac{1}{4} \), as can be shown by integrating over the possible positions or via combinatorial reasoning.

Key Concepts

• Random points on a circle and geometry of triangles.
• Probabilistic method: converting geometric constraints into arc-length inequalities.
• Connections to order statistics and uniform distributions.

Summary Table

Detailed Explanations and Interview Insights

How to Approach Quant Research Interview Questions

Quantitative research interviews, especially at elite firms like Jane Street and SIG, are designed to test not just your mathematical knowledge, but your problem-solving process and communication. Here are some tips and strategies:

• Clarify and Structure: Restate the problem in your own words to clarify the setup and assumptions. Break the problem into manageable parts.
• Use Symmetry and Intuition: Look for patterns, symmetry, or invariances that can simplify calculations, especially in probability and combinatorics problems.
• Conditional Probability: When a problem involves random selection and observation (like the colored cube), carefully define the conditioning event and count all relevant cases.
• Complementary Counting: Sometimes it's easier to count the complement of the desired event, as in the dice matching problem.
• Order Statistics: For stick-breaking or circle-point problems, understanding the distribution of sorted random variables is key.
• Geometric Probability: Translate geometric constraints (angles, arcs, lengths) into inequalities involving random variables.
• Communicate Clearly: Even if you get stuck, narrate your reasoning. Interviewers value a structured thought process.

Common Quant Interview Concepts

• Conditional and joint distributions
• Expectation and tail integration
• Combinatorial enumeration and symmetry
• Bayes’ theorem applications
• Random geometric arrangements (circle, stick, cube)
• Expected value and risk/reward calculations

Code Examples: Simulating the Problems

For practice and intuition, you can simulate these problems using Python. This helps validate the analytical results and build intuition about the distributions involved.

1. Simulating the 3x3x3 Cube Problem

# Enumerate all cubes and faces for the colored cube problem

def cube_faces():
    types = {'corner': (3, 8), 'edge': (2, 12), 'face': (1, 6)} # faces per cube, number of cubes
    total_green_faces = sum(f * n for (f,n) in types.values())
    edge_green_faces = types['edge'][0] * types['edge'][1]
    probability = edge_green_faces / total_green_faces
    return probability

print("Probability edge cube given green face:", cube_faces())

2. Simulating the Dice Game

def simulate_dice_game(n_sim=1000000):
    wins = 0
    for _ in range(n_sim):
        p1 = np.random.randint(1,7,2)
        p2 = np.random.randint(1,7,2)
        if any(d1 == d2 for d1 in p1 for d2 in p2):
            wins += 1
    return wins / n_sim

print("Simulated probability of winning:", simulate_dice_game())

3. Simulating the Circle Problem

def simulate_circle(n_sim=1000000):
    count = 0
    for _ in range(n_sim):
        points = np.sort(np.random.uniform(0, 1, 3))
        arcs = np.array([points[1] - points[0], points[2] - points[1], 1 - points[2] + points[0]])
        if np.max(arcs) < 0.5:
            count += 1
    return count / n_sim

print("Simulated probability triangle contains center:", simulate_circle())

Conclusion

Quantitative research interview questions at SIG, Jane Street, and similar top trading firms are designed to probe your mathematical intuition, combinatorial reasoning, and ability to communicate complex solutions. By working through classic problems such as stick-breaking, colored cubes, dice games, and random points on a circle, you develop not only the technical skills but also the structured problem-solving approach that top firms value. Practice both analytical solutions and simulations to gain confidence and deepen your understanding.

Remember, in quant interviews, the journey—the logical reasoning and clear communication—is as important as the destination. Good luck with your preparation!

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